Well, Nixon didn't take power until 1969. It would be LBJ who instituted the initial bombing halt and started talks in Paris that May. These were steps that no possible successor could reverse, and Johnson probably intended it to be that way, much like Nixon's own intentions with his foreign policy initiatives, particularly with China.

Something to note is that this had all happened before in a different form with public opinion: Eisenhower promised in 1952 to end the Korean War effectively to win election due to frustration with the stalemate in Korea, which was part of what Nixon was trying to emulate in his campaign strategy regarding Vietnam. US culture had drastically changed for the worse (regarding this) since then-not only Vietnam was arguably worse in some ways than Korea in terms of bloodshed, there was chaos in the streets. But that wasn't the only calculus behind changing policies.

Some of it was brutal military reality: the casualty rates, the rising tensions in the army. The strategy of the Johnson Administration was to fight a war of attrition with the Vietnamese, something that not only was a dubious prospect given the realities of American culture, and specifically American culture in the 60s, but grossly underestimated the North Vietnamese capacity for pain tolerance. The credibility of the White House and their claims that victory was around the corner were gone, and Nixon knew it.

Another was the fact that the Vietnam War has basically bogged down American attention to one corner of the world at a time where China was an implacably militant rogue state, the Middle East was simmering in the aftermath of the Six Day War, US-NATO relations were at an all time low, and the Soviet Union had reached nuclear parity with the USA, successfully squashing the Prague Spring without any American response. Pathetic two-bits like North Korea challenged the US with impunity by 1968 with the Pueblo seizure and the DMZ firefights, as they knew that the US couldn't do anything because of Vietnam. It was going to be hard for Nixon or Kissinger to get better relations with the USSR and China, things they cared about a lot more than who ruled in Saigon in the long run, if the war continued at the level it did in 1968 perpetually. Thus, it was in the interests of the United States to find a way to end US ground involvement in an acceptable manner.

Finally, the war was straining America's finances. You couldn't pay for all the new Great Society programs and the war at the same time in the long run.

The point is, the wisdom of committing 550,000 troops the ground had become moot by 1969. No matter what you think of the way they tried to get out of Vietnam, neither Richard Nixon nor Henry Kissinger questioned the desirability of eventual disengagement after Tet, anymore than Barack Obama questioned the desirability of disengagement in 2008. Nixon intentionally only ruled out Goldwater/Reagan style escalation and McCarthy style surrender during his 1968 campaign. Nixon put it best in May of 1969 to General Abrams: "In Saigon the tendency is to fight the war to victory. It has to be kept in mind, but you and I know it won’t happen—it is impossible." He already largely said the same thing to his campaign writers the year before. For all the flaws in Nixon's Vietnam strategy, it is simply untrue to argue that he didn't want to end the war. Of the 20,000 soldiers killed during the Nixon years, 12,000 were killed in 1969, 9,000 in the first six months alone: a reflection of the time that it takes for any new policy to take effect (Vietnamization wasn't even announced until the Midway meeting with Thieu) and the Johnson era strategy of attrition warfare. By the time of the Silent Majority speech, Nixon could point to the fact that no American soldier had been killed in a whole 24 hour cycle, something that had not been true since 1966.

Sources:

Jeffrey Kimball: Nixon's Vietnam War

Richard Reeves: Nixon-Alone in the White House

Walter Isaacson: Kissinger

Jussi Hanhimaki: The Flawed Architect

Or really, just about any Nixon or Kissinger biographer would suffice.